|Abstract||The local critical parameter $\lambda_s$ of continuous-time branching random walks is completely understood and can be computed as a function of the reproduction rates. On the other hand, only for some classes of branching random walks it is known that the global critical parameter $\lambda_w$ is a certain function of the reproduction rates, which we denote by $ 1/K_w$. We provide here new sufficient conditions which guarantee that the global critical parameter equals $ 1/K_w$. This result extends previously known results for branching random walks on multigraphs and general branching random walks.
We show that these sufficient conditions are satisfied by periodic tree-like branching random walks.
We also discuss the critical parameter and the critical behaviour of continuous-time branching processes in varying environment.
So far, only examples where $\lambda_w=1/K_w$ were known; here we
provide an example where $\lambda_w>1/K_w$.|